Problem: Let $m=x^2+3$. Which equation is equivalent to $(x^2+3)^2+7x^2+21=-10$ in terms of $m$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $m^2-7m+31=0$ (Choice B) B $m^2+7m+31=0$ (Choice C) C $m^2-7m+10=0$ (Choice D) D $m^2+7m+10=0$
Solution: We are asked to rewrite the equation in terms of $m$, where ${m}={x^2+3}$. In order to do this, we need to find all of the places where the expression ${x^2+3}$ shows up in the equation, and then substitute ${m}$ wherever we see them! For instance, note that $7x^2+21=7({x^2+3})$. This means that we can rewrite the equation as: $(x^2+3)^2+7x^2+21=-10$ $({x^2+3})^2+7({x^2+3})=-10$ [What if I don't see this factorization?] Now we can substitute ${m}={x^2+3}$ : $({m})^2+7({m})=-10$ Finally, let's manipulate this expression so that it shares the same form as the answer choices: ${m}^2+7{m}+10=0$ In conclusion, $m^2+7m+10=0$ is equivalent to the given equation when $m=x^2+3$.